Decomposition of Euclidean Nearly K\"ahler submanifolds

We study the foliation space of complex and invariant (by torsion of intrinsic Hermitian connection) umbilic distribution on an isometric immersion from a nearly K\"ahler manifold $M$ into the Euclidean space. Under suitable conditions this leaf space is nearly K\"ahler and $M$ can be decomposed into a product of this leaf space and a 6-dimensional locally homogeneous nearly K\"ahler manifold.